Energy partition and attenuation of Lg waves by numerical simulations using screen propagators

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1 Physics of the Earth and Planetary Interiors Energy partition and attenuation of Lg waves by numerical simulations using screen propagators Ru-Shan Wu ), Shengwen Jin, Xiao-Bi Xie Institute of Tectonics, Modeling and Imaging Laboratory, UniÕersity of California at Santa Cruz, Santa Cruz, CA 95064, USA Received 1 January 1998; accepted 1 August 1999 Abstract Energy partition and attenuation of Lg waves in complex crustal waveguides with both large-scale structures and small-scale random heterogeneities are studied by numerical simulations. A newly developed screen propagator method Ž half-space generalized screen propagators. is tested and applied to this problem. The screen method is two to three orders of magnitude faster than finite difference Ž FD. method and uses much less internal memory. The method has no numerical dispersion and can easily incorporate various Q models into the codes. After analyzing different attenuation mechanisms, this paper is concentrated in simulating the leakage attenuation of Lg waves caused by forward large-angle scattering from random heterogeneities, which scatters the guided waves out of the trapped modes and leaking into the mantle. In addition to energy attenuation curves, variations of angular spectra of Lg waves along the path are also shown to give insight on the energy partition and scattering effects. The curve of equivalent Q for leakage attenuation as a function of normalized scale length Ž ka. of the random heterogeneities agrees well with the scattering theory. The comparisons of the method with wavenumber integration and FD method, and the results of the numerical simulations demonstrate the validity and capacity of the screen propagator method in studying Lg attenuation. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Lg wave; Finite difference; Q model 1. Introduction The importance of small-scale random heterogeneities to seismic wave propagation is well-known. There are extensive publications on this subject in seismology. However, the role of random heterogeneities in Lg excitation, propagation, attenuation and blockage is still unclear due to the complexity of ) Corresponding author. Tel.: q ; fax: q address: wrs@es.ucsc.edu Ž R.-S. Wu.. the problem. The theory of wave propagation in unbounded random media has been well-developed. Radiative transfer theory Ž Chandrasekhar, has been introduced into seismology to treat the energy attenuation, coda decay, separation of intrinsic and scattering attenuations ŽWu, 1985; Wu and Aki, 1988; Zeng, 1991, 1993; Zeng et al., 1991; Sato 1994a,b, 1995; for more references, see the book by Sato and Fehler, For crustal structures, i.e., a single layer over a half space, some simple theoretical solutions based on the energy flux model ŽFran- kel and Wennerberg, 1987., which has a basic as r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S

2 228 R.-S. Wu et al.rphysics of the Earth and Planetary Interiors sumption of the homogeneous distribution of coda energy behind the direct wavefront, have been derived for the teleseismic observations ŽLangston, 1989; Korn, In the derivation, a plane wave incidence is assumed and the coda energy is taken as uniformly distributed in the crust. However, for waves in complex crustal waveguides with random heterogeneities, the theoretical difficulties are overwhelming, and no analytical tools are currently available for performing realistic calculations. Numerical simulation is an attractive alternative to the theory. Some finite difference Ž FD. simulations have been conducted Že.g., Frankel, 1989; Frankel and Clayton, 1986; Xie and Lay, 1994; Jih, Limited by the computation power, however, the FD results are often obtained only for short distances or low frequencies. Liu and Wu Ž have done some numerical simulations using the phase screen method, but the simulations are limited to unbounded media. The development of the half-space Generalized Screen Propagator Ž GSP. method ŽWu et al., 1996, 1997, enables us to simulate long-range, high-frequency waves propagation in highly complex crustal waveguides including random small-scale heterogeneities. The generalized screen method is based on the one-way wave equation and the one-return approximation. The one-way wave propagator GSP neglects backscattered waves, but correctly handles all the forward multiple-scattering effects, e.g., focusingr defocusing, diffraction, interference, and conversion between different wave types. The method is two to three orders of magnitude faster than the FD method for medium size problems. The screen method has been successfully used in forward modeling ŽWu, 1994, 1996; Wu and Huang, 1995; Xie and Wu 1995, 1996; Wild and Hudson, and as backpropagators for seismic wave imagingrmigration in both acoustic and elastic media Že.g., Stoffa et al., 1990; Wu and Xie, 1994; Huang and Wu, 1996; Huang and Fehler, 1998, 1999; Jin et al., In the crustal waveguide environment, major wave energy is carried by forward-propagating waves, including forward-scattered waves; therefore, the neglect of backscattered waves in the propagation will not change the main features of regional waves in most cases. By neglecting reverberation in the theory, the method becomes a forward-marching algorithm in which the next step of propagation depends only on the present values of the wavefield in a transverse cross-section and the heterogeneities between the two cross-sections. The saving of computing time and storage is enormous. In the paper by Wu et al. Ž 1999., a half-space GSP has been introduced to accommodate the free-surface boundary condition to treat the SH wave propagation in complex crustal waveguides. At present, it is probably the only realistic simulation method for highfrequency Ž up to 25 Hz., long-range Ž G 1000 km. Lg wave propagation. In Section 2, the theory and method will be summarized. In Section 3, the Lg energy partition and attenuation will be discussed and simulated using the half-space GSP method. Finally a conclusion is given in Section A brief summary of the theory and method For a 2D SH problem, only the y-component of the displacement field, noted as u, exists. With the perturbation method, the medium and the wave field are decomposed into: rsr qdr ; msm qdm; usu 0 qu, 0 0 where r0 and m0 are the density and shear rigidity of the background medium, d r and dm are the corresponding perturbations, u 0 is the primary field and U is the scattered field. The SH wave equation in the frequency domain can be rewritten as: m= 0 uqv r0usy v druq= dm= u. 1 For each step of the marching algorithm under the forward-scattering approximation, the total field at x1 is calculated as the sum of the primary field which is the field propagating in the half space from x X to x 1, and the scattered field caused by the heterogeneities in the thin slab between x X and x 1. The thickness of the slab should be made thin enough to ensure the validity of the local Born approximation. Under this condition, the Green s function can be approximated by the homogeneous half-space Green s function. The latter can be obtained by the image method. The stress should vanish at the free

3 R.-S. Wu et al.rphysics of the Earth and Planetary Interiors surface z s 0. Under these conditions, the scattered field can be calculated by Ž Wu et al., 1999.: UŽ x 1, K z. sur Ž x 1, K z. qum Ž x 1, K z., x1 k ig Ž x 1 yx. U Ž x, K. sik d xe C Ž z. u Ž z., H r 1 z r 0 x X g H x 1 ig Ž x 1 yx. Um x 1, K z sik d xe C m z Ex u0 z X x K z m z 0 g 5 ½ yi S Ž z. E u Ž z., Ž 2. where dr Ž r. dmž r. rž r. s, mž r. s, r m ( 2 2 gs k yk z, E 1 E Exs, Ezs. ik Ex ik Ez In the above equations, Cw fž z.x and SwfŽ z.x are the cosine and sine transforms: H ` C fž z. s d z2cosž K z. fž z., 0 ` H S fž z. s d z2sinž K z. fž z., Ž 3. 0 and u, E u and E u can be calculated by: 0 x 0 z 0 1 ` X X X X ik z z ig Ž xyx. X X 0 z 0 z H u Ž x, z. s d K e e u Ž x, K. 2p y` X X y1 ig Ž xyx. X X sc e u Ž x, K., Ž 4. 0 z X X X g y1 ig Ž xyx. X X x 0 0 z E u Ž x, z. sc e u Ž x, K. k X X X K y1 ig Ž xyx. z X X z 0 0 z E u Ž x, z. si S e u Ž x, K.. Ž 5. k The above equations are the general wide-angle formulation. When the energy of crustal guided waves is carried mainly by small-angle waves Žwith respect to the horizontal direction., the phase screen approximation can be invoked to simplify the theory and calculations. Summing up the primary and scattered fields and invoking the Rytov transform result z z in the dual-domain phase space expression of phase screen propagator for this case: X ig Ž x 1 yx. 2 iks s Ž z. X Ž 1 z. 0 u x, K fe C e u x, z, where e 2 iks sž z. is the phase delay operator with 1 x1 H SsŽ z. s d x rž x, z. y mž x, z. fd x sž z. 2 X x Ž 6., where Ž z. s is the average S-wave slowness pertur- bation over the thin-slab at depth z: s Ž z. s H d x, x yx 1 x1 s x, z ys0 X X 1 x s0 with sž x, z. s1ržõž x, z.. and D xsž x yx X. 1 is the thin-slab thickness Implementation procedure for the half-space phase screen propagator Under the phase screen approximation, the heterogeneous half space is represented by a series of half screens embedded in the homogeneous background half space. The wave propagates between screens in the wavenumber domain and interacts with the phase screens in the space domain. The interaction is only a phase-delay operator Žmultiplication in space domain.. The procedure can be summarized as follows: 1. Cosine-transform the incident field at the starting plane into wavenumber domain and free propagate to the screen; 2. Inverse cosine-transform the incident field into space domain and interact with the shear slowness screen Ž phase screen. to get the transmitted field; 3. Repeat the propagation and interaction screenby-screen to the boundary of the model space Treatment of the Moho discontinuity The Moho discontinuity can be treated in two ways. One is to put the impedance boundary conditions in the formulation, the other is to treat the parameter changes as perturbations and therefore be

4 230 R.-S. Wu et al.rphysics of the Earth and Planetary Interiors incorporated into the screen interaction. The former has the advantage of computational efficiency. The latter has the flexibility of handling irregular interfaces. In this paper, we adopt the latter approach and check the validity of perturbation approach for the Moho discontinuity by the wavenumber integration and FD algorithms. For guided waves, or crustal waves with critical or post-critical reflections, the related mantle waves are nearly horizontal. Therefore, the screen approximation is quite accurate in this case. The excellent agreements of the method with the wavenumber integration for flat Moho, and with the FD method for irregular Moho demonstrated the validity of this approach ŽWu et al., Energy partition and attenuation in crusts with random heterogeneities In heterogeneous crustal waveguides, the upper boundary is the free surface, which is a perfect reflector. The lower boundary of the waveguide is the Moho discontinuity. For waves incident on the Moho discontinuity, part of the energy will leak into the upper mantle. However, for waves incident on the Moho with post-critical angles, total reflections occur and all the energy are reflected and trapped in the waveguide. Generally speaking, the guided wave energy can be expressed as: H E s < už K. < 2 d K, Ž 7. g z z K z-k c where K z is the wavenumber in the z-direction, namely the transverse wavenumber, and K c is the critical wavenumber. Scattering processes can redistribute the energy in wavenumber domain, causing the leak of trapped energy into the upper mantle. In addition to the leakage loss, the guided waves suffer also the anelastic loss and backscattering loss. Assuming a homogeneous mantle and neglecting reverberation in the x-direction, the energy balance after propagating a short distance d x in the x-direction is: EgŽ xqd x. segž x. yeaž x. yebž x. yelž x., Ž 8. where Eg is the energy of guided crustal waves; E a, energy lost due to absorption Ž anelastic loss.; E b, energy lost due to backscattering by random heterogeneities; E l, energy lost due to leakage to the mantle caused by heterogeneities. In terms of different attenuation coefficients, it can be written as: d E rd xsywh qh qh xe Ž x. syh E, Ž 9. g a b l g g g where h sž E re. rd x, h sž E re. a a g b b g rd x, and hl sž E re. l g rd x are the apparent attenuation coeffi- cients for guided crustal waves. Equivalently: h svq y1 sv Q y1 qq y1 qq y1, Ž 10. g g a b l where the Qs are the corresponding apparent quality factors. The leakage loss is the scattering loss due to the redistribution of Lg angular spectra. It is caused dominantly by large-angle forward scattering, and therefore is orders of magnitude larger than the backscattering loss, i.e., hl4h b. In the following, we will concentrate on the analysis of leakage loss of guided waves. For the leakage analysis, the angular spectrum representation or the energy distribution vs. Fig. 1. Comparison of synthetic seismograms along the surface calculated by the screen method Ž thick lines. and reflectivity method Ž thin lines. for a flat crustal model Ž 32 km thick.. The point source depth is 2 km and the source function is a Ricker wavelet with dominant frequency of 1.0 Hz.

5 R.-S. Wu et al.rphysics of the Earth and Planetary Interiors propagation angle Ž or vertical slowness. will be very useful and can show clearly which part of the energy would be trapped in the waveguide and which part of the energy would leak into the mantle. In first order approximation, the anelastic attenuation Ž intrinsic attenuation. is additive to the leakage loss, so that we can calculate and analyze the latter independently. For the Lg RMS amplitude attenuation, one more attenuation mechanism is involved: ˆy1 y1 y1 y1 y1 bgsvqg sv Qa qqb qql qq d, 11 where Q y1 d is the equivalent Q of diffusion loss, which represents the amplitude decrease of Lg due to the transfer of coherent energy into incoherent en- ergy Lg coda by random heterogeneities Energy partition for a reference waõeguide model First, let us test the screen propagator for a flat crustal waveguide. In Fig. 1, we show the comparison of synthetic seismograms generated by the screen method Ž thick lines. with those calculated by a reflectivity method Ž thin lines.. The crust has a thickness of 32 km and a shear wave velocity of 3.5 kmrs. The mantle beneath the crust has a shear velocity of 4.5 kmrs. The point source depth is 2 km Fig. 2. Energy distribution for a uniform crustal model. The crust thickness is 32 km and the source depth is 2 km. From top to bottom, crustal structure; energy angular spectra vs. distance; and energy attenuation vs. distance.

6 232 R.-S. Wu et al.rphysics of the Earth and Planetary Interiors and the source function is a Ricker wavelet with a dominant frequency of 1.0 Hz. The results from the screen method agree very well with that from the reflectivity method, which is considered very accurate for flat layered media. The only exception is for near-vertical reflections at very small epicentral distances where the screen method has low accuracy due to extremely large scattering angles. However, since regional seismograms are usually recorded at large distances, this limitation does not pose any real Fig. 3. Energy angular spectra vs. distance for different frequencies for a uniform crust. From top to bottom, the frequencies are 2.0, 1.5, 1.0 and 0.5 Hz, respectively.

7 R.-S. Wu et al.rphysics of the Earth and Planetary Interiors problem for its applications. Fig. 2 shows the angular spectra along the propagation path and the energy attenuation for the flat crust, hereafter referred to as the reference model. Shown in the upper panel of Fig. 2 is the waveguide structure. In the middle is the energy distribution vs. the vertical slowness and distance up to 600 km. The vertical coordinate is the normalized vertical slowness K zrk, corresponding to the cosine of incident angles Žor sine of the grazing angles.. Note that zero vertical slowness means horizontal propagation. The frequency range is from 0.6 to 1.9 Hz. At the initial stage, there is considerable portion of energy with large vertical slowness, i.e., with steep angles. After multiple reflections, energy with larger vertical slowness is depleted due to the leakage to the mantle, leaving the energy with small vertical slowness, i.e., the guided waves, propagating in the waveguide. Shown in the bottom panel is the wave energy vs. the distance. The energy is calculated from synthetic seismograms on the free surface. It can be seen that after passing 100 km or more, the energy is kept basically constant, which means that the trapped mode has been formed. Fig. 3 shows the energy distribution vs. the vertical slowness as a function of distance for the reference model for individual frequency components. From top to bottom are plots for frequencies 2.0, 1.5, 1.0 and 0.5 Hz, respectively. The energy distribution shows some patterns in the slowness distance domain. These result from the interference of waves from the source and the reflected waves from the free surface and the Moho. After a certain distance, the patterns are relatively stable, i.e., certain modes have formed in the waveguide and they carry the energy to a long distance. Irregular waveguide structures Ž large scale. and random heterogeneities Ž small scale. can affect the propagating mode, i.e., the patterns of energy distributions WaÕeguides with random heterogeneities Small-scale heterogeneities are known to widely exist in the crust. These small-scale structures can be simulated by random velocity fluctuations on a background velocity and by random interface undulations. The following models will be used to test how Fig. 4. Comparison of energy attenuation curves calculated by screen propagator method and FD method. Shown in the top panel is the velocity structure including random heterogeneities and in the lower panel are the energy attenuation curves.

8 234 R.-S. Wu et al.rphysics of the Earth and Planetary Interiors these small-scale structures affect Lg energy partition and attenuation. First, we check the validity of the screen method in dealing with the guided wave attenuation by comparing its result with that from a FD method. The upper panel of Fig. 4 gives a crust model with 5% RMS velocity perturbation. To shorten the computation time, here we use a 16-kmthin crust model. The lower panel shows the comparison of relative attenuation curves between the two methods. The solid line is from a fourth order FD method and the dotted line is from screen propagator method. The two results agree reasonably well. This proves the validity of the half-space screen propagator applied to the energy transfer problems. For this test model, the FD calculation has D xsd zs0.125 km, Dts0.015 s resulting in a CPU time of 58 h on a SUN SPARC-4 work station, while the screen method has D xsd zs0.25 km, Dts0.1 s, and a CPU time 0.5 h on the same machine. Both calculations have f0 s 1 Hz. For the screen method, the cutoff frequency fmax s 2 Hz. Shown in Fig. 5 is a crustal waveguide similar to the reference model except a 5% RMS random velocity perturbation in the crust. The perturbation has an exponential correlation function with horizontal and vertical characteristic scales Ž correlation lengths. of 5.0 and 3.0 km, respectively. Compared with Fig. 2, the distinct feature for this case is the continuous Fig. 5. Energy distribution for a crust model with 5% RMS velocity perturbations. The horizontal and vertical characteristic lengths Ž correlation length. of the random medium are 5.0 and 3.0 km, respectively. From top to bottom, crustal structure; energy angular spectra vs. distance; and relative energy attenuation vs. distance.

9 R.-S. Wu et al.rphysics of the Earth and Planetary Interiors energy repartition, moving from small Ž grazing. -angle waves to large-angle waves due to scattering by small-scale heterogeneities. In the middle panel of Fig. 5, more energy with the pre-critical wavenumbers Ž large vertical slowness. can be seen comparing to that for the reference model in Fig. 2. This portion of energy Ž large angle waves. tends to leak into the mantle and cause Lg wave energy attenuation. This is shown in the bottom panel of the figure, where the dotted line is for the reference model and solid line Fig. 6. Angular spectra vs. distance for a crust model with 5% RMS velocity perturbations for different frequencies. From top to bottom, the frequencies are 2.0, 1.5, 1.0 and 0.5 Hz, respectively.

10 236 R.-S. Wu et al.rphysics of the Earth and Planetary Interiors is for the random waveguide. Fig. 6 is the energy distribution vs. cosine of incident angle as a function of distance for individual frequency components. Comparing with those in Fig. 3 for the reference model, we can see that many interference patterns for guided modes are destroyed. This is especially clear for higher frequency components. The energy partition and leakage attenuation due to scattering is strongly dependent on the characteristic scale Žor equivalently, the spatial power spectrum. of the random heterogeneities. Fig. 7 gives the attenuation curves for different characteristic scales. The upper panel is the attenuation curve of total energy, which is the energy contained in the whole seismogram recorded on the surface. The thin solid line is for kas1, the thick solid line is for kas10, and the dashed line is for the reference model. We see that for the reference model, the total energy remains constant beyond critical distance, which serves as a checking point for the numerical simulations. The middle panel gives the coherent Lg energy, which is calculated using waves within the Lg window Žgroup velocity between 3.7 and 3.2 kmrs. vs. distance. Again, the thin, thick and dashed lines are for kas1, Fig. 7. Total energy attenuation Ž top panel., and windowed Lg energy attenuation with group velocities kmrs Ž middle panel. vs. y1 distance for kas1 thin lines and kas10 thick lines. The bottom panel shows the equivalent Q for leakage attenuation vs. the normalized scale length ka. The dashed line is for the reference model of a homogeneous crust.

11 R.-S. Wu et al.rphysics of the Earth and Planetary Interiors Table 1 Apparent quality factor Ql for energy leakage attenuation of Lg waves vs. the normalized scale length Ž ka. of random heterogeneities ka Ql kas10 and the reference model. In both measurements, ka s 1 cases are associated with stronger attenuation than kas10 cases. We see also that the coherent Lg energy corresponding to the peak amplitude suffers more attenuations than the total energy. This is due to the extra attenuation, the diffusion loss which scatters the waves out of the Lg window and transfers them into incoherent waves Ž Lg coda.. However, in these numerical simulations, there is no intrinsic attenuation and leakage attenuation dominates; the difference between the coherent energy attenuation and the total energy attenuation is relatively small. In the bottom panel, we plot the curve of apparent inverse quality factor for leakage attenuy1 ation Q vs. the normalized scale length Ž ka. l of random heterogeneities, where k s 2prl with l being the wavelength of the dominant frequency, and a the correlation length. The Ql values are also listed in Table 1. In the Q calculations, first, the l Fig. 8. Energy distribution for a model with crustal necking. From top to bottom, crustal structure; energy angular spectra vs. distance; and energy attenuation vs. distance. The solid line is for the necking model, and the dashed line for the reference model of a homogeneous crust.

12 238 R.-S. Wu et al.rphysics of the Earth and Planetary Interiors total energy attenuation curves in the range of km are fitted into straight lines by least-square fitting. The attenuation coefficients thus obtained are then used to calculate the equivalent Qs. Since no intrinsic Ž anelastic. attenuation exists in the model and no backscattering is involved, the attenuation is assumed to be totally caused by the leakage loss due to scattering. From the curve and table, we see that Q y1 l reaches its peak at kaf1.5 2 and keeps flat until kaf8. This is a feature of large-angle forward scattering dominance. For backscattering, the maximum scattering Q y1 is around kaf1 and decreases rapidly at ka) 1 for exponential correlation functions; while for large-angle forescattering, the plateau is quite wide after kas1 ŽWu, 1982; Frankel and Clayton, The numerical simulations agree well with the scattering theory. The values of the equivalent Q in Table 1 Ž for f s1 Hz. 0 are comparable with some observations ŽXie, 1993; Xie and Mitchell, This suggests that the leakage attenuation caused by small-scale random heterogeneities may be responsible and even the dominant mechanism for some observed Lg attenuations and blockages. Fig. 9. Energy distribution for a model with crustal thickening. From top to bottom, crustal structure; energy angular spectra vs. distance; and energy attenuation vs. distance. The solid line is for the thickening model, and the dashed line for the reference model of a homogeneous crust.

13 R.-S. Wu et al.rphysics of the Earth and Planetary Interiors Crusts with both large-scale structures and random heterogeneities When both large-scale structures and random heterogeneities exit, wave propagation in the crustal waveguide becomes more complicated. Due to the strong coupling between the effects of complex crustal structures and random heterogeneities, the energy leakage of the combined structures may be stronger than the sum of the individual leakages. The GSP method has the advantage of effectively simulating wave interaction with both large- and smallscale heterogeneities. In this section, we show some examples to demonstrate the capability of the simulation method. Systematic study of the energy attenuation in different types of waveguides will be conducted in the future. To see the influences of large-scale structures to energy leakage, we first show the cases of crust thinning Ž neck type. and thickening Ž belly type.. Fig. 8 shows the neck-type crustal model Ž top panel., angular spectrum variation Ž middle panel., and the energy attenuation Ž bottom panel. along the propagation path. On the neck, the crust thickness decreases from 30 to 20 km. Due to the existence of slopes, especially the uphill slope, the total reflection condition has been destroyed in such sections and part of the wave energy will leak to the mantle. This can be seen from the spectral broadening near the uphill slope. The leakage caused by the downhill slope is Fig. 10. A heterogeneous crustal model representing a mountain root with small-scale random heterogeneities top panel. The comparison between synthetic seismograms with and without random heterogeneities is shown on the middle and bottom panels, respectively.

14 240 R.-S. Wu et al.rphysics of the Earth and Planetary Interiors far less severe. These can also be seen from the snapshots of wave fields propagating through the relevant sections Ž Wu et al., The temporary increase near the uphill slope in the energy attenuation curve is caused by the focusing effect of that structure. The model in Fig. 9 has a crustal thicken- Fig. 11. Comparison between snapshots for waves passing through a mountain root with or without random heterogeneities, shown on A and B, respectively.

15 R.-S. Wu et al.rphysics of the Earth and Planetary Interiors ing. Similar spectral broadening and energy leakage can be observed. As an example of combined multi-scale structures, Fig. 10 shows a heterogeneous crustal model representing a mountain root with small-scale random heterogeneities. The top panel shows the velocity model, and the synthetic seismograms with and without random heterogeneities are shown on the middle and bottom panels, respectively. The heterogeneities have an exponential correlation function, with the scale length a s a s 1.6 km Ž x z in horizontal and vertical directions, respectively.. The RMS velocity perturbation is 5%. The dominant frequency of the source function is 2 Hz. Fig. 11A Ž top two panels. and B Ž bottom two panels. show the comparison between snapshots for waves passing through the mountain root with and without random heterogeneities, respectively. We see that random heterogeneities significantly increase the leakage of waves to the mantle and the complexity of the waveforms. Fig. 12 shows the spectral variation Ž middle panel. and energy attenuation Ž bottom panel. for the random mountain root model Ž top panel.. Compared with Fig. 9, the random heterogeneities reduce the amplitude fluctuations caused by the focusing effect of the slope. The spectral broadening occurs more evenly within the random region. The attenuation measured within or immediately after the Fig. 12. Energy distribution for the bellying crustal model with small-scale random heterogeneities. From top to bottom panel, crustal structure; energy angular spectra vs. distance; and energy attenuation vs. distance. The solid line is for the model with random heterogeneities, and the dashed line for the reference model of a homogeneous crust.

16 242 R.-S. Wu et al.rphysics of the Earth and Planetary Interiors random region has significant increase. The overall attenuation has also increased compared with that of Fig. 9. It has been shown that the crustal random heterogeneities are highly anisotropic in scale length and nonuniformly distributed in depths ŽHolliger and Levander, 1992; Levander and Holliger, 1992; Wu et al., The influences of the random heterogeneities with different stochastic characteristics will be explored systematically in the future work. 4. Conclusion For medium size 2D Lg problems, the screen method Ž half-space GSP. is two to three orders of magnitude faster than FD method. The GSP needs only to store 2D data arrays for each step instead of 3D volume data, leading to huge memory savings. The other advantage of the GSP method is that it has no numerical dispersion since the transversal Laplacian is calculated by the Fourier method. Being a frequency domain method, it is also easy to incorporate various Q models into simulations. Therefore, the method can be used to study the effects of scattering and anelasticity on long-range, highfrequency Lg wave propagation and attenuation for complex crustal waveguides including small-scale random heterogeneities. Attenuations of the total energy and coherent Lg energy Ž within the Lg window. have been studied by numerical simulations using the screen method. In addition to the energy attenuation curve, variation of the angular spectrum of Lg waves along the propagation path can also offer some insight on the energy partition and scattering effects. From the simulations for waveguides filled with random heterogeneities, it is shown that the leakage attenuation caused by forward scattering by the random heterogeneities, which scatter the wave out of the trapped modes, is significant and may become the dominant attenuation mechanism in some regions. For the random crustal models in the paper, with 5% velocity perturbation and a dominant frequency f0 s 1 Hz, the equivalent Q of leakage attenuation for different scale lengths of heterogeneities ranges from 300 to 900 for ka s The peak attenuation occurs near ka f 1.5. This agrees well with the scattering theory. The results of numerical simulations given in this paper are mainly for the purpose of demonstration of the capability of the method. The crustal structure used in this paper is simplified or specially designed. The results of scattering and attenuation from the simulations may differ from the real earth structures. Systematic numerical simulations will be conducted and reported in future publications to study the influences of different attenuation mechanisms on Lg attenuation and blockage for realistic crustal structures. Acknowledgements The helpful discussion with T. Lay is greatly appreciated. The work was supported by The Department of Energy and The Office of Non-Proliferation and National Security through contract F K-0016, administered by the Phillips Laboratory of the Air Force and by DSWA through contract DSWA The facility support from the W.M. Keck Foundation is also acknowledged. Contribution no. 412 of the Institute of Tectonics, University of California, Santa Cruz. References Chandrasekhar, S., Radiation Transfer. Dover, New York. Frankel, A., A review of numerical experiments on seismic wave scattering. In: Wu, R.S., Aki, K. Ž Eds.., Scattering and Attenuation of Seismic Waves II. Birkhauser, Berlin, pp Frankel, A., Clayton, R.W., Finite difference simulations of seismic scattering: implications for propagation of short-period seismic waves in the crust and models of crustal heterogeneity. J. Geophys. Res. 91, Frankel, A., Wennerberg, L., Energy-flux model of seismic coda: separation of scattering and intrinsic attenuation. Bull. Seismol. Soc. Am. 77, Holliger, K., Levander, A.R., A stochastic view of lower crustal fabric based on evidence from the Ivrea zone. Geophys. Res. Lett. 19, Huang, L.J., Fehler, M.C., Accuracy analysis of the split-step Fourier propagator: implications for seismic modeling and migration. Bull. Seismol. Soc. Am. 88, Huang, L.J., Fehler, M.C., Local Born and local Rytov Fourier migration method. Geophysics, Ž in press.. Huang, L.J., Wu, R.S., D prestack depth migration with acoustic pseudo-screen propagators. In: Mathematical Methods in Geophysical Imaging IV Vol SPIE, pp

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